We aim at studying various structures and algorithms in other spaces
than Rd, from a computational geometry viewpoint.
Proposing algorithms operating in such spaces requires a prior deep study of the
mathematical properties of the objects considered, which raises new
fundamental and difficult questions that we want to tackle.

Methodology.
A key characteristic of the project is its
interdisciplinarity: it gathers approaches, knowledge, and tools in
mathematics and computer science.
A mathematical study of the considered objects will be
performed, together with the design of algorithms when
applicable. Algorithms will be analyzed both in theory and in practice
after prototype implementations. In the long term,
implementations should be improved whenever
it makes sense to target longer-term integrations into CGAL, in order
to disseminate our results to end-users.

Challenges. The global difficulty of this project is
intrinsic to the differences between the classical Euclidean spaces
and the spaces that we consider (to spot only one concrete example:
translations in the hyperbolic plane do not commute). This prevents
known algorithms of computational geometry from naturally extending to
other spaces. The mathematical difficulties are real and new ideas
will be necessary.
The challenge is to provide new mathematical foundations for
computational geometry, in order to allow the emergence of algorithms that will
be both efficient in theory and effective in practice.

Visits to/from Groningen are partially supported by the University of Groningen.
Visits to/from Luxembourg are partially supported by the
University of Luxembourg. Since January 2018, those visits, as well as
visits between INRIA and LIGM, fall under
the umbrella of the project SoS, co-funded by
ANR and FNR.